Abstract

In this paper we investigate the ground-state properties and related quantum phase transitions for the two-component Bose-Einstein condensate in a single-mode optical cavity. Apart from the usual normal and superradiant phases, multi-stable macroscopic quantum states are realized by means of the spin-coherent-state variational method. We demonstrate analytically the stimulated radiation from a collective state of atomic population inversion, which does not exist in the normal Dicke model with single-component atoms. It is also revealed that the stimulated radiation can be generated only from one component of atoms and the other remains in the ordinary superradiant state. However, the order of superradiant and stimulated-radiation states is interchangeable between two components of atoms by tuning the relative atom-field couplings and the frequency detuning as well.

© 2017 Optical Society of America

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2016 (1)

Z. M. Wang, J. L. Lian, J.-Q. Liang, Y. M. Yu, and W. M. Liu, “Collapse of the superradiant phase and multiple quantum phase transitions for Bose-Einstein condensates in an optomechanical cavity,” Phys. Rev. A 93, 033630 (2016).
[Crossref]

2014 (4)

J. T. Fan, Z. W. Yang, Y. W. Zhang, J. Ma, G. Chen, and S. T. Jia, “Hidden continuous symmetry and Nambu-Goldstone mode in a two-mode Dicke model,” Phys. Rev. A,  89, 023812 (2014).
[Crossref]

Y. C. Zhang, X. F. Zhou, G. C. Guo, X. X. Zhou, H. Pu, and Z. W. Zhou, “Two-component polariton condensate in an optical microcavity,” Phys. Rev. A 89, 053624 (2014).
[Crossref]

X. Q. Zhao, N. Liu, and J.-Q. Liang, “Nonlinear atom-photon-interaction-induced population inversion and inverted quantum phase transition of Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 90, 023622 (2014).
[Crossref]

A. B. Bhattacherjee, “Non-equilibrium dynamical phases of two-Atom Dicke model,” Phys. Lett. A 378, 3244 (2014).
[Crossref]

2013 (8)

J. L. Lian, N. Liu, J.-Q Liang, G. Chen, and S. T. Jia, “Ground-state properties of a Bose-Einstein condensate in an optomechanical cavity,” Phys. Rev. A 88, 043820 (2013).
[Crossref]

N. Liu, J. D. Li, and J. -Q. Liang, “Nonequilibrium quantum phase transition of Bose-Einstein condensates in an optical cavity,” Phys. Rev. A 87, 053623 (2013).
[Crossref]

A. Wickenbroc, M. Hemmerling, G. R. M. Robb, C. Emary, and F. Renzoni, “Collective strong coupling in multi-mode cavity QED,” Phys. Rev. A 87, 043817 (2013).
[Crossref]

M. Buchhold, P. Strack, S. Sachdev, and S. Diehl, “Dicke-model quantum spin and photon glass in optical cavities: Nonequilibrium theory and experimental signatures,” Phys. Rev. A 87, 063622 (2013).
[Crossref]

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553 (2013).
[Crossref]

R. Puebla, A. Relaño, and J. Retamosa, “Excited-state phase transition leading to symmetry-breaking steady states in the Dicke model,” Phys. Rev. A 87, 023819 (2013).
[Crossref]

D. J. Egger and F. K. Wilhelm, “Multimode circuit quantum electrodynamics with hybrid metamaterial transmission lines,” Phys. Rev. Lett. 111, 163601 (2013).
[Crossref] [PubMed]

C.-P. Yang, Q.-P. Su, S.-B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
[Crossref]

2012 (3)

H. Cao and L. B. Fu, “Quantum phase transition and dynamics induced by atom-pair tunnelling of Bose-Einstein condensates in a double-well potential,” Eur. Phys. J. D 66, 97 (2012).
[Crossref]

M. J. Bhaseen, J. Mayoh, B. D. Simons, and J. Keeling, “Dynamics of nonequilibrium Dicke models,” Phys. Rev. A,  85, 013817 (2012).
[Crossref]

J. L. Lian, Y. W. Zhang, and J.-Q Liang, “Macroscopic quantum states and quantum phase transition in the Dicke model,” Chin. Phys. Lett. 29, 060302 (2012).
[Crossref]

2011 (14)

O. Castaños, E. Nahmad-Achar, R. López-Peñna, and J. G. Hirsch, “No singularities in observables at the phase transition in the Dicke model,” Phys. Rev. A 83, 051601 (2011).
[Crossref]

M. Eto, K. Kasamatsu, M. Nitta, H. Takeuchi, and M. Tsubota, “Interaction of half-quantized vortices in two-component Bose-Einstein condensates,” Phys. Rev. A 83, 063603 (2011).
[Crossref]

C. A. Regal and K. W. Lehnert, “From cavity electromechanics to cavity optomechanics,” J. Phys.: Conf. Ser. 264, 012025 (2011).

Y. Dong, J. W. Ye, and H. Pu, “Multistability in an optomechanical system with a two-component Bose-Einstein condensate,” Phys. Rev. A 83, 031608 (2011).
[Crossref]

K Sasaki, N. Suzuki, and H. Saito, “Capillary instability in a two-component Bose-Einstein condensate,” Phys. Rev. A 83, 053606 (2011).
[Crossref]

S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, “Frustration and glassiness in spin models with cavity-mediated interactions,” Phys. Rev. Lett. 107, 277201 (2011).
[Crossref]

P. Strack and S. Sachdev, “Dicke quantum spin glass of atoms and photons,” Phys. Rev. Lett. 107, 277202 (2011).
[Crossref]

N. Liu, J. L. Lian, J. Ma, L. T. Xiao, G. Chen, J-Q. Liang, and S. T. Jia, “Light-shift-induced quantum phase transitions of a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 83, 033601 (2011).
[Crossref]

P. Nataf and C. Ciuti, “Protected quantum computation with multiple resonators in ultrastrong coupling circuit QED,” Phys. Rev. Lett. 107, 190402 (2011).
[Crossref] [PubMed]

J. Q. You and F. Nori, “Atomic physics and quantum optics using superconducting circuits,” Nature (London) 474, 589 (2011).
[Crossref]

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

Y. Eto, A. Noguchi, P. Zhang, M. Ueda, and M. Kozuma, “Projective measurement of a single nuclear spin qubit by using two-mode cavity QED,” Phys. Rev. Lett. 106, 160501 (2011).
[Crossref] [PubMed]

M. Mariantoni, H. Wang, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Photon shell game in three-resonator circuit quantum electrodynamics,” Nat. Phys. 7, 287 (2011).
[Crossref]

K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, “Exploring symmetry breaking at the Dicke quantum phase transition,” Phys. Rev. Lett. 107, 140402 (2011).
[Crossref] [PubMed]

2010 (8)

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature (London) 464, 1301 (2010).
[Crossref]

J. Larson and J. -P. Martikainen, “Ultracold atoms in a cavity-mediated double-well system,” Phys. Rev. A 82, 033606 (2010).
[Crossref]

L. Zhou, H. Pu, H. Y. Ling, K. Zhang, and W. P. Zhang, “Spin dynamics and domain formation of a spinor Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 81, 063641 (2010).
[Crossref]

A. O. Silver, M. Hohenadler, M. J. Bhaseen, and B. D. Simons, “Bose-Hubbard models coupled to cavity light fields,” Phys. Rev. A 81, 023617 (2010).
[Crossref]

G. Szirmai, D. Nagy, and P. Domokos, “Quantum noise of a Bose-Einstein condensate in an optical cavity, correlations, and entanglement,” Phys. Rev. A 81, 043639 (2010).
[Crossref]

J. Larson, “Analog of the spin-orbit-induced anomalous Hall effect with quantized radiation,” Phys. Rev. A 81, 051803 (2010).
[Crossref]

J. Keeling, M. J. Bhaseen, and B. D. Simons, “Collective dynamics of Bose-Einstein condensates in optical cavities,” Phys. Rev. Lett. 105, 043001 (2010).
[Crossref] [PubMed]

M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, “Quantum optomechanics—throwing a glance,” J. Opt. Soc. Am. B 27, A189 (2010).
[Crossref]

2009 (10)

K. Sasaki, N. Suzuki, D. Akamatsu, and H. Saito, “Rayleigh-Taylor instability and mushroom-pattern formation in a two-component Bose-Einstein condensate,” Phys. Rev. A 80, 063611 (2009).
[Crossref]

J.-Q. Liang, J.-L. Liu, W.-D. Li, and Z.-J. Li, “Atom-pair tunneling and quantum phase transition in the strong-interaction regime,” Phys. Rev. A 79, 033617 (2009).
[Crossref]

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

I. Favero and K. Karrai, “optomechanics of deformable optical cavities,” Nature Photonics 3, 201 (2009).
[Crossref]

S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, “Emergent crystallinity and frustration with Bose-Einstein condensates in multimode cavities,” Nat. Phys. 5, 845 (2009).
[Crossref]

J. Larson and S. Levin, “Effective abelian and non-abelian gauge potentials in cavity QED,” Phys. Rev. Lett. 103, 013602 (2009).
[Crossref] [PubMed]

D. G. Norris, E. J. Cahoon, and L. A. Orozco, “Atom detection in a two-mode optical cavity with intermediate coupling: Autocorrelation studies,” Phys. Rev. A 80, 043830 (2009).
[Crossref]

M. L. Terraciano, R. Olson Knell, D. G. Norris, J. Jing, A. Fernández, and L. A. Orozco, “Photon burst detection of single atoms in an optical cavity,” Nat. Phys. 5, 480 (2009).
[Crossref]

G. Szirmai, D. Nagy, and P. Domokos, “Excess noise depletion of a Bose-Einstein Condensate in an optical cavity,” Phys. Rev. Lett. 102, 080401 (2009).
[Crossref] [PubMed]

M. J. Bhaseen, M. Hohenadler, A. O. Silver, and B. D. Simons, “Polaritons and Pairing Phenomena in Bose-Hubbard Mixtures,” Phys. Rev. Lett. 102, 135301 (2009).
[Crossref] [PubMed]

2008 (7)

J. Larson, B. Damski, G. Morigi, and M. Lewenstein, “Mott-insulator states of ultracold atoms in optical resonators,” Phys. Rev. Lett. 100, 050401 (2008).
[Crossref] [PubMed]

S. Morrison and A. S. Parkins, “Dynamical quantum phase transitions in the dissipative Lipkin-Meshkov-Glick model with proposed realization in optical cavity QED,” Phys. Rev. Lett. 100, 040403 (2008).
[Crossref] [PubMed]

J. M. Zhang, W. M. Liu, and D. L. Zhou, “Mean-field dynamics of a Bose Josephson junction in an optical cavity,” Phys. Rev. A 78, 043618 (2008).
[Crossref]

G. Chen, X. G. Wang, J. -Q. Liang, and Z. D. Wang, “Exotic quantum phase transitions in a Bose-Einstein condensate coupled to an optical cavity,” Phys. Rev. A 78, 023634 (2008).
[Crossref]

M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, “Two-resonator circuit quantum electrodynamics: A superconducting quantum switch,” Phys. Rev. B 78, 104508 (2008).
[Crossref]

T. J. Kippenberg and K. J. Vahala, “Cavity optomechanics: Back-action at the mesoscale,” Science 321, 1172 (2008)
[Crossref] [PubMed]

B. V. Schaeybroeck, “Interface tension of Bose-Einstein condensates,” Phys. Rev. A 78, 023624 (2008).
[Crossref]

2007 (4)

F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A 75, 013804 (2007).
[Crossref]

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272 (2007).
[Crossref]

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature (London) 450, 268 (2007).
[Crossref]

D. Tolkunov and D. Solenov, “Quantum phase transition in the multimode Dicke model,” Phys. Rev. B 75, 024402 (2007).
[Crossref]

2006 (1)

G. Chen, J. Q. Li, and J.-Q. Liang, “Critical property of the geometric phase in the Dicke model,” Phys. Rev. A 74, 054101 (2006).
[Crossref]

2004 (1)

Z.-D. Chen, J.-Q. Liang, S.-Q. Shen, and W. -F. Xie, “Dynamics and Berry phase of two-species Bose-Einstein condensates,” Phys. Rev. A 69, 023611 (2004).
[Crossref]

2003 (3)

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: The Dicke model,” Phys. Rev. Lett. 90, 044101 (2003).
[Crossref] [PubMed]

A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, “Many-particle entanglement in two-component Bose-Einstein condensates,” Phys. Rev. A 67, 013607 (2003).
[Crossref]

C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E 67, 066203 (2003).
[Crossref]

2002 (1)

R. A. Barankov, “Boundary of two mixed Bose-Einstein condensates,” Phys. Rev. A 66, 013612 (2002).
[Crossref]

2001 (2)

A. Søensen, L.-M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature(London) 409, 63 (2001).
[Crossref]

P. Horak and H. Ritsch, “Dissipative dynamics of Bose condensates in optical cavities,” Phys. Rev. A 63, 023603 (2001).
[Crossref]

1999 (2)

D. Gordon and C. M. Savage, “Creating macroscopic quantum superpositions with Bose-Einstein condensates,” Phys. Rev. A 59, 4623 (1999).
[Crossref]

R. F. Fox, “Generalized coherent states,” Phys. Rev. A 59, 3241 (1999).
[Crossref]

1998 (2)

E. Timmermans, “Phase separation of Bose-Einstein condensates,” Phys. Rev. Lett. 81, 5718 (1998).
[Crossref]

H. Pu and N.P. Bigelow, “Properties of two-species Bose condensates,” Phys. Rev. Lett. 80, 1130 (1998).
[Crossref]

1997 (1)

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of interference between two Bose condensates,” Science 275, 637 (1997).
[Crossref] [PubMed]

1996 (1)

Y.-Z. Lai, J.-Q. Liang, H. J. W. Müller-Kirsten, and J. G. Zhou, “Time-dependent quantum systems and the invariant Hermitian operator,” Phys. Rev. A 53, 3691 (1996).
[Crossref] [PubMed]

1990 (1)

E. Layton, Y. H. Huang, and S. I. Chu, “Cyclic quantum evolution and Aharonov-Anantlan geometric phases in SU(2) spin-coherent states,” Phys. Rev. A 41, 42 (1990).
[Crossref] [PubMed]

1978 (1)

R. Gilmore and L.M. Narducci, “Relation between the equilibrium and nonequilibrium critical properties of the Dicke model,” Phys. Rev. A 17, 1747 (1978).
[Crossref]

1977 (1)

B. V. Thompson, “A canonical transformation theory of the generalized Dicke model,” J. Phys. A 10, 89 (1977).
[Crossref]

1973 (2)

Y. K. Wang and F. T. Hioe, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A,  7, 831 (1973).
[Crossref]

F. T. Hioe, “Phase transitions in some generalized Dicke models of superradiance,” Phys. Rev. A 8, 1440 (1973).
[Crossref]

1954 (1)

R. H. Dicke, “Coherence in Spontaneous Radiation Processes,” Phys. Rev. 93, 99 (1954).
[Crossref]

Akamatsu, D.

K. Sasaki, N. Suzuki, D. Akamatsu, and H. Saito, “Rayleigh-Taylor instability and mushroom-pattern formation in a two-component Bose-Einstein condensate,” Phys. Rev. A 80, 063611 (2009).
[Crossref]

Andrews, M. R.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of interference between two Bose condensates,” Science 275, 637 (1997).
[Crossref] [PubMed]

Aspelmeyer, M.

Barankov, R. A.

R. A. Barankov, “Boundary of two mixed Bose-Einstein condensates,” Phys. Rev. A 66, 013612 (2002).
[Crossref]

Baumann, K.

K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, “Exploring symmetry breaking at the Dicke quantum phase transition,” Phys. Rev. Lett. 107, 140402 (2011).
[Crossref] [PubMed]

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature (London) 464, 1301 (2010).
[Crossref]

Bhaseen, M. J.

M. J. Bhaseen, J. Mayoh, B. D. Simons, and J. Keeling, “Dynamics of nonequilibrium Dicke models,” Phys. Rev. A,  85, 013817 (2012).
[Crossref]

J. Keeling, M. J. Bhaseen, and B. D. Simons, “Collective dynamics of Bose-Einstein condensates in optical cavities,” Phys. Rev. Lett. 105, 043001 (2010).
[Crossref] [PubMed]

A. O. Silver, M. Hohenadler, M. J. Bhaseen, and B. D. Simons, “Bose-Hubbard models coupled to cavity light fields,” Phys. Rev. A 81, 023617 (2010).
[Crossref]

M. J. Bhaseen, M. Hohenadler, A. O. Silver, and B. D. Simons, “Polaritons and Pairing Phenomena in Bose-Hubbard Mixtures,” Phys. Rev. Lett. 102, 135301 (2009).
[Crossref] [PubMed]

Bhattacherjee, A. B.

A. B. Bhattacherjee, “Non-equilibrium dynamical phases of two-Atom Dicke model,” Phys. Lett. A 378, 3244 (2014).
[Crossref]

Bialczak, R. C.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

M. Mariantoni, H. Wang, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Photon shell game in three-resonator circuit quantum electrodynamics,” Nat. Phys. 7, 287 (2011).
[Crossref]

Bigelow, N.P.

H. Pu and N.P. Bigelow, “Properties of two-species Bose condensates,” Phys. Rev. Lett. 80, 1130 (1998).
[Crossref]

Bourdel, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature (London) 450, 268 (2007).
[Crossref]

Brandes, T.

C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E 67, 066203 (2003).
[Crossref]

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: The Dicke model,” Phys. Rev. Lett. 90, 044101 (2003).
[Crossref] [PubMed]

Brennecke, F.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553 (2013).
[Crossref]

K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, “Exploring symmetry breaking at the Dicke quantum phase transition,” Phys. Rev. Lett. 107, 140402 (2011).
[Crossref] [PubMed]

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature (London) 464, 1301 (2010).
[Crossref]

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature (London) 450, 268 (2007).
[Crossref]

Buchhold, M.

M. Buchhold, P. Strack, S. Sachdev, and S. Diehl, “Dicke-model quantum spin and photon glass in optical cavities: Nonequilibrium theory and experimental signatures,” Phys. Rev. A 87, 063622 (2013).
[Crossref]

Cahoon, E. J.

D. G. Norris, E. J. Cahoon, and L. A. Orozco, “Atom detection in a two-mode optical cavity with intermediate coupling: Autocorrelation studies,” Phys. Rev. A 80, 043830 (2009).
[Crossref]

Cao, H.

H. Cao and L. B. Fu, “Quantum phase transition and dynamics induced by atom-pair tunnelling of Bose-Einstein condensates in a double-well potential,” Eur. Phys. J. D 66, 97 (2012).
[Crossref]

Carmichael, H. J.

F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A 75, 013804 (2007).
[Crossref]

Castaños, O.

O. Castaños, E. Nahmad-Achar, R. López-Peñna, and J. G. Hirsch, “No singularities in observables at the phase transition in the Dicke model,” Phys. Rev. A 83, 051601 (2011).
[Crossref]

Chen, G.

J. T. Fan, Z. W. Yang, Y. W. Zhang, J. Ma, G. Chen, and S. T. Jia, “Hidden continuous symmetry and Nambu-Goldstone mode in a two-mode Dicke model,” Phys. Rev. A,  89, 023812 (2014).
[Crossref]

J. L. Lian, N. Liu, J.-Q Liang, G. Chen, and S. T. Jia, “Ground-state properties of a Bose-Einstein condensate in an optomechanical cavity,” Phys. Rev. A 88, 043820 (2013).
[Crossref]

N. Liu, J. L. Lian, J. Ma, L. T. Xiao, G. Chen, J-Q. Liang, and S. T. Jia, “Light-shift-induced quantum phase transitions of a Bose-Einstein condensate in an optical cavity,” Phys. Rev. A 83, 033601 (2011).
[Crossref]

G. Chen, X. G. Wang, J. -Q. Liang, and Z. D. Wang, “Exotic quantum phase transitions in a Bose-Einstein condensate coupled to an optical cavity,” Phys. Rev. A 78, 023634 (2008).
[Crossref]

G. Chen, J. Q. Li, and J.-Q. Liang, “Critical property of the geometric phase in the Dicke model,” Phys. Rev. A 74, 054101 (2006).
[Crossref]

Chen, Z.-D.

Z.-D. Chen, J.-Q. Liang, S.-Q. Shen, and W. -F. Xie, “Dynamics and Berry phase of two-species Bose-Einstein condensates,” Phys. Rev. A 69, 023611 (2004).
[Crossref]

Chu, S. I.

E. Layton, Y. H. Huang, and S. I. Chu, “Cyclic quantum evolution and Aharonov-Anantlan geometric phases in SU(2) spin-coherent states,” Phys. Rev. A 41, 42 (1990).
[Crossref] [PubMed]

Cirac, J. I.

A. Micheli, D. Jaksch, J. I. Cirac, and P. Zoller, “Many-particle entanglement in two-component Bose-Einstein condensates,” Phys. Rev. A 67, 013607 (2003).
[Crossref]

A. Søensen, L.-M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature(London) 409, 63 (2001).
[Crossref]

Ciuti, C.

P. Nataf and C. Ciuti, “Protected quantum computation with multiple resonators in ultrastrong coupling circuit QED,” Phys. Rev. Lett. 107, 190402 (2011).
[Crossref] [PubMed]

Cleland, A. N.

H. Wang, M. Mariantoni, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Deterministic entanglement of photons in two superconducting microwave resonators,” Phys. Rev. Lett. 106, 060401 (2011).
[Crossref] [PubMed]

M. Mariantoni, H. Wang, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, A. D. O’Connell, D. Sank, M. Weides, J. Wenner, T. Yamamoto, Y. Yin, J. Zhao, J. M. Martinis, and A. N. Cleland, “Photon shell game in three-resonator circuit quantum electrodynamics,” Nat. Phys. 7, 287 (2011).
[Crossref]

Colombe, Y.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272 (2007).
[Crossref]

Damski, B.

J. Larson, B. Damski, G. Morigi, and M. Lewenstein, “Mott-insulator states of ultracold atoms in optical resonators,” Phys. Rev. Lett. 100, 050401 (2008).
[Crossref] [PubMed]

Deppe, F.

M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, “Two-resonator circuit quantum electrodynamics: A superconducting quantum switch,” Phys. Rev. B 78, 104508 (2008).
[Crossref]

Dicke, R. H.

R. H. Dicke, “Coherence in Spontaneous Radiation Processes,” Phys. Rev. 93, 99 (1954).
[Crossref]

Diehl, S.

M. Buchhold, P. Strack, S. Sachdev, and S. Diehl, “Dicke-model quantum spin and photon glass in optical cavities: Nonequilibrium theory and experimental signatures,” Phys. Rev. A 87, 063622 (2013).
[Crossref]

Dimer, F.

F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A 75, 013804 (2007).
[Crossref]

Domokos, P.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553 (2013).
[Crossref]

G. Szirmai, D. Nagy, and P. Domokos, “Quantum noise of a Bose-Einstein condensate in an optical cavity, correlations, and entanglement,” Phys. Rev. A 81, 043639 (2010).
[Crossref]

G. Szirmai, D. Nagy, and P. Domokos, “Excess noise depletion of a Bose-Einstein Condensate in an optical cavity,” Phys. Rev. Lett. 102, 080401 (2009).
[Crossref] [PubMed]

Dong, Y.

Y. Dong, J. W. Ye, and H. Pu, “Multistability in an optomechanical system with a two-component Bose-Einstein condensate,” Phys. Rev. A 83, 031608 (2011).
[Crossref]

Donner, T.

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature (London) 450, 268 (2007).
[Crossref]

Duan, L.-M.

A. Søensen, L.-M. Duan, J. I. Cirac, and P. Zoller, “Many-particle entanglement with Bose-Einstein condensates,” Nature(London) 409, 63 (2001).
[Crossref]

Dubois, G.

Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, “Strong atom-field coupling for Bose-Einstein condensates in an optical cavity on a chip,” Nature (London) 450, 272 (2007).
[Crossref]

Durfee, D. S.

M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle, “Observation of interference between two Bose condensates,” Science 275, 637 (1997).
[Crossref] [PubMed]

Egger, D. J.

D. J. Egger and F. K. Wilhelm, “Multimode circuit quantum electrodynamics with hybrid metamaterial transmission lines,” Phys. Rev. Lett. 111, 163601 (2013).
[Crossref] [PubMed]

Emary, C.

A. Wickenbroc, M. Hemmerling, G. R. M. Robb, C. Emary, and F. Renzoni, “Collective strong coupling in multi-mode cavity QED,” Phys. Rev. A 87, 043817 (2013).
[Crossref]

C. Emary and T. Brandes, “Quantum chaos triggered by precursors of a quantum phase transition: The Dicke model,” Phys. Rev. Lett. 90, 044101 (2003).
[Crossref] [PubMed]

C. Emary and T. Brandes, “Chaos and the quantum phase transition in the Dicke model,” Phys. Rev. E 67, 066203 (2003).
[Crossref]

Esslinger, T.

H. Ritsch, P. Domokos, F. Brennecke, and T. Esslinger, “Cold atoms in cavity-generated dynamical optical potentials,” Rev. Mod. Phys. 85, 553 (2013).
[Crossref]

K. Baumann, R. Mottl, F. Brennecke, and T. Esslinger, “Exploring symmetry breaking at the Dicke quantum phase transition,” Phys. Rev. Lett. 107, 140402 (2011).
[Crossref] [PubMed]

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature (London) 464, 1301 (2010).
[Crossref]

F. Brennecke, T. Donner, S. Ritter, T. Bourdel, M. Köhl, and T. Esslinger, “Cavity QED with a Bose-Einstein condensate,” Nature (London) 450, 268 (2007).
[Crossref]

Estienne, B.

F. Dimer, B. Estienne, A. S. Parkins, and H. J. Carmichael, “Proposed realization of the Dicke-model quantum phase transition in an optical cavity QED system,” Phys. Rev. A 75, 013804 (2007).
[Crossref]

Eto, M.

M. Eto, K. Kasamatsu, M. Nitta, H. Takeuchi, and M. Tsubota, “Interaction of half-quantized vortices in two-component Bose-Einstein condensates,” Phys. Rev. A 83, 063603 (2011).
[Crossref]

Eto, Y.

Y. Eto, A. Noguchi, P. Zhang, M. Ueda, and M. Kozuma, “Projective measurement of a single nuclear spin qubit by using two-mode cavity QED,” Phys. Rev. Lett. 106, 160501 (2011).
[Crossref] [PubMed]

Fan, J. T.

J. T. Fan, Z. W. Yang, Y. W. Zhang, J. Ma, G. Chen, and S. T. Jia, “Hidden continuous symmetry and Nambu-Goldstone mode in a two-mode Dicke model,” Phys. Rev. A,  89, 023812 (2014).
[Crossref]

Favero, I.

I. Favero and K. Karrai, “optomechanics of deformable optical cavities,” Nature Photonics 3, 201 (2009).
[Crossref]

Fernández, A.

M. L. Terraciano, R. Olson Knell, D. G. Norris, J. Jing, A. Fernández, and L. A. Orozco, “Photon burst detection of single atoms in an optical cavity,” Nat. Phys. 5, 480 (2009).
[Crossref]

Fox, R. F.

R. F. Fox, “Generalized coherent states,” Phys. Rev. A 59, 3241 (1999).
[Crossref]

Fu, L. B.

H. Cao and L. B. Fu, “Quantum phase transition and dynamics induced by atom-pair tunnelling of Bose-Einstein condensates in a double-well potential,” Eur. Phys. J. D 66, 97 (2012).
[Crossref]

Gilmore, R.

R. Gilmore and L.M. Narducci, “Relation between the equilibrium and nonequilibrium critical properties of the Dicke model,” Phys. Rev. A 17, 1747 (1978).
[Crossref]

Girvin, S. M.

F. Marquardt and S. M. Girvin, “Optomechanics,” Physics 2, 40 (2009).
[Crossref]

Goldbart, P. M.

S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, “Frustration and glassiness in spin models with cavity-mediated interactions,” Phys. Rev. Lett. 107, 277201 (2011).
[Crossref]

S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, “Emergent crystallinity and frustration with Bose-Einstein condensates in multimode cavities,” Nat. Phys. 5, 845 (2009).
[Crossref]

Gopalakrishnan, S.

S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, “Frustration and glassiness in spin models with cavity-mediated interactions,” Phys. Rev. Lett. 107, 277201 (2011).
[Crossref]

S. Gopalakrishnan, B. L. Lev, and P. M. Goldbart, “Emergent crystallinity and frustration with Bose-Einstein condensates in multimode cavities,” Nat. Phys. 5, 845 (2009).
[Crossref]

Gordon, D.

D. Gordon and C. M. Savage, “Creating macroscopic quantum superpositions with Bose-Einstein condensates,” Phys. Rev. A 59, 4623 (1999).
[Crossref]

Gröblacher, S.

Gross, R.

M. Mariantoni, F. Deppe, A. Marx, R. Gross, F. K. Wilhelm, and E. Solano, “Two-resonator circuit quantum electrodynamics: A superconducting quantum switch,” Phys. Rev. B 78, 104508 (2008).
[Crossref]

Guerlin, C.

K. Baumann, C. Guerlin, F. Brennecke, and T. Esslinger, “Dicke quantum phase transition with a superfluid gas in an optical cavity,” Nature (London) 464, 1301 (2010).
[Crossref]

Guo, G. C.

Y. C. Zhang, X. F. Zhou, G. C. Guo, X. X. Zhou, H. Pu, and Z. W. Zhou, “Two-component polariton condensate in an optical microcavity,” Phys. Rev. A 89, 053624 (2014).
[Crossref]

Hammerer, K.

Han, S. Y.

C.-P. Yang, Q.-P. Su, S.-B. Zheng, and S. Y. Han, “Generating entanglement between microwave photons and qubits in multiple cavities coupled by a superconducting qutrit,” Phys. Rev. A 87, 022320 (2013).
[Crossref]

Hemmerling, M.

A. Wickenbroc, M. Hemmerling, G. R. M. Robb, C. Emary, and F. Renzoni, “Collective strong coupling in multi-mode cavity QED,” Phys. Rev. A 87, 043817 (2013).
[Crossref]

Hioe, F. T.

Y. K. Wang and F. T. Hioe, “Phase transition in the Dicke model of superradiance,” Phys. Rev. A,  7, 831 (1973).
[Crossref]

F. T. Hioe, “Phase transitions in some generalized Dicke models of superradiance,” Phys. Rev. A 8, 1440 (1973).
[Crossref]

Hirsch, J. G.

O. Castaños, E. Nahmad-Achar, R. López-Peñna, and J. G. Hirsch, “No singularities in observables at the phase transition in the Dicke model,” Phys. Rev. A 83, 051601 (2011).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

Schematic diagram for two ensembles of ultracold atoms (blue and green) with transition frequencies ω1, ω2 in an optical cavity of frequency ω.

Fig. 2
Fig. 2

Graphical solutions of the extremum equation pk (γsk) = 0 for k = ⇊ (black line), k (olive line), and k =⇅ (blue line) with g1/ω = 0.6 and g2/ω = 0.2 (a1), 0.4 (a2), 0.7 (a3), 0.9 (a4). The corresponding average energy curves ε are plotted in the lower panel (b1–b4). γ ¯ 2 = γ 2 / N denotes the mean photon number.

Fig. 3
Fig. 3

Phase diagram in the resonance condition ω1 = ω2 = ω. The notations NPts(N, N,N) and NPts (N, N,N) mean the NP with triple states, in which N is the ground state. SPco (S, N, N) [SPco (S, N,N)] means the SP characterized by the ground state S, which coexists with N (N) and N. SPco (S, S,N) [SPco (S, S,N)] is also the coexisting SP, in which the first excited-state is a superradiant state S (S).

Fig. 4
Fig. 4

Phase diagram in g-Δ space with the atom-photon coupling parameter δ = 0 (a), δ = 0.5 (b), and δ = −0.5 (c). The boundary line, which separates the regions with different first-excited-states (N, S and N, S), moves upward and downward respectively for δ = 0.5 (b), −0.5 (c).

Fig. 5
Fig. 5

Variations of the average photon number np (a), atom population imbalance Δna (b), and average energy ε (c) with respect to the coupling constant g = g1 = g2 in the atom-field frequency detuning Δ = 0.6 (1) and Δ = −0.6 (2).

Fig. 6
Fig. 6

The average photon number np (a), atomic population Δna (b), and average energy ε (c) curves for the imbalance parameter δ = −0.5 (1), δ = 0.5 (2) in the resonance condition Δ = 0.0. The stimulated radiation shifts from one component to the other by adjusting the relative coupling constants.

Equations (51)

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H = ω a a + l = 1 , 2 ω l J l z + l = 1 , 2 g l N l ( a + a ) ( J l + + J l ) .
H s p ( α ) = α | H | α = ω α * α + l = 1 , 2 ω l J l z + l = 1 , 2 g l N l ( α * + α ) ( J l + + J l ) ,
| ± n l = R ( n l ) | s , ± s l ,
R ( n l ) = e θ l 2 ( J l + J e i φ l J l e i φ l ) .
| ψ s = | ± n 1 | ± n 2 ,
H s p ( α ) | ψ s = E ( α ) | ψ s .
| ψ s = U | ± s 1 | ± s 2 ,
U = R ( n 1 ) R ( n 2 ) ,
H ˜ s p ( α ) | ± s 1 | ± s 2 = E ( α ) | ± s 1 | ± s 2 ,
H ˜ s p ( α ) = U H s p ( α ) U .
J ˜ l z = J l z cos θ l + 1 2 sin θ l ( J l + e i φ l + J l e i φ l ) , J ˜ l + = J l + cos 2 θ l 2 J l e 2 i φ l sin 2 θ l 2 J l z e i φ l sin θ l , J ˜ l = J l cos 2 θ l 2 J l + e 2 i φ l sin 2 θ l 2 J l z e i φ l sin φ l .
ω l 2 e i φ l sin θ l + g ˜ l α ( cos 2 θ l 2 e 2 i φ l sin 2 θ l 2 ) = 0 , ω l 2 e i φ l sin θ l + g ˜ l α ( cos 2 θ l 2 e 2 i φ l sin 2 θ l 2 ) = 0 ,
E ( α ) = ω | α | 2 ± l = 1 , 2 N l 2 A l ( α , θ l , φ l ) ,
A l ( α , θ l , φ l ) = ω l cos θ l g ˜ l α ( e i φ l + e i φ l ) sin θ l ,
| ψ = | α | ψ s ,
α = γ e i ϕ .
E ω ( γ ) = γ 2 ± l = 1 , 2 , N l 2 ( ω l ω ) 2 + 16 γ 2 N l ( g l ω ) 2 .
E ( γ ) ω = γ 2 l = 1 , 2 N l 2 ( ω l ω ) 2 + 16 γ 2 N l ( g l ω ) 2 .
ω 1 = ω Δ , ω 2 = ω + Δ .
ε = E ( γ ) N ω
ε γ = 2 γ p ( γ ) = 0 ,
p ( γ ) = 1 l = 1 , 2 4 g l 2 ω 2 F l ( γ ) ,
F l ( γ ) = ( ω l ω ) 2 + 32 ( g l ω ) 2 γ 2 N .
2 ( ε   ( γ   2 = 0 ) γ 2 = 2 [ 1 4 ω ( g 1 2 ω 1 + g 2 2 ω 2 ) ] ,
g 1 , c 2 ω 1 + g 2 , c 2 ω 2 = ω 4 .
g 1 2 ω 1 + g 2 2 ω 2 < ω 4 ,
ε = γ 2 N 1 4 [ F 1 ( γ ) F 2 ( γ ) ]
p ( γ ) = 1 4 ω 2 [ g 1 2 F 1 ( γ ) g 2 2 F 2 ( γ ) ] ,
2 ( ε ( γ 2 = 0 ) ) γ 2 = 2 N [ 1 4 ω ( g 1 2 ω 1 g 2 2 ω 2 ) ]
g 1 2 ω 1 g 2 2 ω 2 < ω 4 .
ε = γ 2 N + 1 4 [ F 1 ( γ ) F 2 ( γ ) ] .
p ( γ ) = 1 + 4 ω 2 ( g 1 2 F 1 ( γ ) g 2 2 F 2 ( γ ) ) .
g 2 2 ω 2 g 1 2 ω 1 < ω 4 .
ε = E ( γ ) ω N = γ 2 N + 1 4 l = 1 , 2 F l ( γ ) .
( ε   ) γ = 2 γ   p   ( γ   ) = 0 ,
p   ( γ   ) = 1 + l = 1 , 2 4 g l 2 ω 2 F l ( γ   ) .
2 ε ( γ 2 = 0 ) γ 2 = 2 N [ 1 + 4 ω ( g 1 2 ω 1 + g 2 2 ω 2 ) ] > 0 ,
p k ( γ s k ) = 0
g 2 = 1 2 1 ( 2 g 1 ω ) 2 , g 2 = 1 2 ( 2 g 1 ω ) 2 + 1 , g 2 = 1 2 ( 2 g 1 ω ) 2 1 .
g 1 = g , g 2 = ( 1 + δ ) g .
g c = 1 2 ( ω 2 Δ 2 ) ω [ 2 ω + ( ω Δ ) ( 2 δ + δ 2 ) ] ,
g c = 1 2 ( ω 2 Δ 2 ) ω [ ( ω + Δ ) ( ω Δ ) ( 1 + δ ) 2 ] = 1 2 ( ω 2 Δ 2 ) ω [ 2 Δ ( ω Δ ) ( 2 δ + δ 2 ) ] ,
g c = 1 2 ( ω 2 Δ 2 ) ω [ ( ω Δ ) ( 1 + δ ) 2 ( ω + Δ ) ] = 1 2 ( ω 2 Δ 2 ) ω [ ( ω Δ ) ( 2 δ + δ 2 ) 2 Δ ] .
n p ( ) = α | a a | α N = { 0 , g < g c , γ 2 N , g > g c . .
Δ n a ( ) = ψ s ( s , s ) | ( J 1 z + J 2 z ) | ψ s ( s , s ) N = 1 4 l = 1 , 2 ω l ω F l ( γ ) ,
Δ n a ( ) = 1 2 ,
ε = { 0.5 , g < g c , γ 2 N 1 4 l = 1 , 2 F l ( γ ) , g > g c . .
n p ( N k ) = 0 ; n p ( S k ) = γ k 2 N .
Δ n a ( N k ) = 0
Δ n a ( S ) = 1 4 ω [ ω 1 F 1 ( γ ) + ω 2 F 2 ( γ ) ] , Δ n a ( S ) = 1 4 ω [ ω 1 F 1 ( γ ) ω 2 F 2 ( γ ) ] .
ε ( N ) = 1 4 ω ( ω 1 + ω 2 )

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